And now I say that a homotopy between two maps induces a chain homotopy between the two chain maps and . And, indeed, if the homotopy is given by a smooth map then we can write , where and are the two boundary inclusions of into the “homotopy cylinder” , and we will work with these inclusions first.

Since , we have chain maps , and we’re going to construct a chain homotopy . That is, for any differential form we will have the equation

Given this, we can write

which shows that is then a chain homotopy from to .

Sometimes the existence of the chain homotopy is referred to as the Poincaré lemma; sometimes it’s the general fact that a homotopy induces the chain homotopy ; sometimes it’s a certain corollary of this fact, which we will get to later. Given my categorical bent, I take it to be the general assertion that we have a 2-functor between the homotopy 2-category and that of chain complexes, chain maps, and chain homotopies.

As a side note: now we can finally understand what the name “chain homotopy” means.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.